Generalized Factorials and Fixed Divisors over Subsets of a Dedekind Domain

Given a subsetXof a Dedekind domainD, and a polynomialF∈D[x], thefixed divisor d(X, F) ofFoverXis defined to be the ideal inDgenerated by the elementsF(a),a∈X. In this paper we derive a simple expression ford(X, F) explicitly in terms of the coefficients ofF, using a generalized notion of "factorial" introduced by the author in a previous paper. WhenX=D=Z, this generalized factorial reduces to the ordinary factorial function; hence we obtain as special cases classic results of Pólya, Gunji, and McQuillan relatingd(Z, F) and the usual factorial function.